1. Let U24 = {z € C|224 = 1}. Define $24 = cos () + i sin (5). %3D (a) List all subgroups of U24 and draw its lattice diagram. (b) Let W = ((§24)20) < U24. What is [U24 : W]? (c) Enumerate all left cosets of W in U24.

Q: 11. Let K be the splitting field of x³ – 2 over Q. (Refer to Example 50.9.) a. Describe the six…

A: please see the next step for solution

Q: 9. Describe the group of the polynomial (x* – 1) e Q[x] over Q.

A:

Q: 12. Describe the group of the polynomial (x* – 5x? + 6) € Q[x] o over Q.

A: please see the next step for solution

Q: 1. Consider H= (i) and K= (-1) the subgroups of Q, the quaternion group. 1) Explain why H is normal…

A: Basic theory regarding normal subgroups from algebra will be used

Q: 10.3.3. Let G Z/4Z x Z/6Z, and let H be the subgroup of G generated by (2, 2). (a) What are the…

A: Given that G=ℤ/4ℤ×ℤ/6ℤ.i.e. G=ℤ4×ℤ6H is a subgroup of G generated by 2,2i.e. H=&lt;2,2&gt; oG=4×6=24

Q: 1) Prove that G = {a + bv2 \a,b € Q and a, b # 0} is a subgroup of (R – (0},.)

A: Subgroup

Q: 2. In this exercise, we will construct a new finite group of order 8 using the quater- nions. The…

A: (a)  Recall  that the Quaternion group is  the group which consists   of the  eight quaternions .…

Q: Compute the indicated values for the indicated homomorphisms. Φ((12)(231)(23)), where Φ: S_3-->Z_2

A:

Q: 6. Describe the left cosets of SL2(R) in GL₂(R). What is the index of SL₂(R) in GL₂(R)?

A: The left coset for a group G with a subgroup H, tis defined by . Since  can be changed; therefore,…

Q: (a) Make addition and multiplication tables for Z/3Z.

A: “Since you have posted multiple questions, we will provide the solution only to the first question…

Q: 4 Let f = (4 5 6) and g = (1984) (2 7 5) (36) be two permutations in S9. (a) Compute fogo f-1, and…

A:

Q: 1. Consider H= (i) and K= (-1) the subgroups of Qg the quaternion group. 4) Explain why K is normal…

A:

Q: 4.6 List all the elements of (Z45,) that are of order 15.

A: (.) Given  group is  ⊕ .(.). The group  ⊕ is an abelian group under addition modulo   and given by ,…

Q: Let f and g be permutations on the set {1, 2, 3, 4, 5, 6, 7}, defined as follows (b) f 1 (c) fgf =…

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Q: 2. Let G = GL(2, R). Prove that the following two subsets of GL(2, R) are subgroups of GL(2, R). (a)…

A: As per the policy we are solving first question only. Please repost the question and specify which…

Q: 12. (Exercise 16.22) Prove the third isomorphism theorem. At each major step in the proof, explain…

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Q: 4. If G = (Z8,) then G has an element of order 5. *True *False

A:

Q: (2) In the group G =Z, let H =(4) and K =(8) %3D (i) (ii) List the elements inG /H, GIK and in H/K ,…

A: G=ℤ24H=0,4,8,12,16,20K=0,8,16

Q: Let A: l²l² be given by 1 2 {x1,5x2,. 5² A(x1, x2,...) = = Find the spectrum o(A). n 2n + 1 In,

A: As per the question we are given a linear transformation A : l2 → l2 and we have to find its…